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2 Precise attitude determination - Precise attitude determination has an application in navigation, geodesy, gravimetry, magnetometry, construction and so on - Carrier phase measurements provided by GPS hardware have potential accuracy of millimeters in linear distance and several advantages compared to inertial navigation systems; this allows using them for attitude determination - Some special properties of carrier phase measurements require a stack of different mathematical methods and computational techniques to be processed instead of one algorithm - Observability and consistency would be checked and evaluated at every stage of calculations - Different methods and its effectiveness could be objectively compared using actual survey data - Precisely known distances between antennas are available as an additional a priori information that could be involved in computational process

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4 Carrier phase measurements - Commonly used carrier phase measurements equation (n is a number of visible satellites, m is number of antennas, N is time span in epochs): p is carrier frequency number, all items are measured in cycles of a carrier The wanted quantity is The rest of components are errors and instrumental noise

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5 Single- and double-differencing - Single-differencing cancels errors due to radio-signal propagation through ionosphere and troposphere, satellite clock errors: - Double-differencing cancels errors due to receiver clock errors; remaining irremovable errors are of less magnitude than others: - Number of measurements is reduced by 2

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6 Attitude determination by processing carrier phase measurements characterization (page 1) - The matter of attitude determination is relative positioning of several antennas (not less than 3) - Differencing is quite natural: the equation of measurement is nearly linear - Precise absolute positioning is not necessary - Additional information (distances between antennas) is available to use

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7 Attitude determination by processing carrier phase measurements characterization (page 2) - When antennas are close to each other (in dozens of meters) errors due to signal propagation through ionosphere and troposphere, linearization and clock errors are completely canceled by double-differencing - Integer ambiguities introduce a significant complexity in estimation; the estimated vector is not observable at every particular epoch; observability is possible when a set of measurements for several epochs is available

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8 Formal statements for different approaches to attitude determination by processing carrier phase measurements (page 1) - Weighted least-squares method (errors in measurements are correlated) - Least-squares method with special restrictions given by known distances between antennas; certain algorithms deliver the solution

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10 Formal statements for different approaches to attitude determination by processing carrier phase measurements (page 3) - Triple-differencing free of integer ambiguities and using special properties of conditionality

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11 Least-squares method (LSM) compared to LSM with restrictions - Least-squares method is well formulated in variety of books - LSM with restrictions is produced as iterative algorithm with linearized restrictions of equation type; least-squares solution is the initial value

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13 Dynamics of observability during constellation evolution - Let H 1 and H 2 be observation matrices for two epochs for m carrier phase measurements containing integer ambiguities - Necessary and sufficient condition for observability is as follows: - The condition states that direction vectors to satellites are not transformed by a single linear operator; furthermore the difference from any that transformation would be enough to satisfy the condition - The more suitable equivalent condition is the m-dimensional basis at the bottom n – m rows of matrices: H 1 =U 1 S 1 V 1 T is singular value decomposition, H 1 =Q 1 R 1 is QR-decomposition

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Field surveys - Field surveys have been conducted several times in Moscow region and once near Tver - Topcon hardware and software - Two antennas - Static and dynamic series - Lengthy series of measurement